Related papers: Lorentz-Invariant Hamiltonian and Riemann Hypothes…

We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen…

This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical…

We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $\zeta(s)$.

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the…

In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two…

We define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of…

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a…

Results of a multipart work are outlined. Use is made therein of the conjunction of the Riemann hypothesis, RH, and hypotheses advanced by the author. Let z(n) be the nth nonreal zero of the Riemann zeta-function with positive imaginary…

The GUE Hypothesis, which concerns the distribution of zeros of the Riemann zeta-function, is used to evaluate some integrals involving the logarithmic derivative of the zeta-function. Some connections are shown between the GUE Hypothesis…

Finite differences of values of the Riemann zeta function at the integers are explored. Such quantities, which occur as coefficients in Newton series representations, have surfaced in works of Maslanka, Coffey, Baez-Duarte, Voros and…

An ensemble of 2 x 2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian Unitary Ensemble found by Wigner. By a re-interpretation of Connes' spectral…

We discuss a possible spectral realization of the Riemann zeros based on the Hamiltonian $H = xp$ perturbed by a term that depends on two potentials, which are related to the Berry-Keating semiclassical constraints. We find perturbatively…

We investigate a dynamical basis for the Riemann hypothesis (RH) that the non-trivial zeros of the Riemann zeta function lie on the critical line x = 1/2. In the process we graphically explore, in as rich a way as possible, the diversity of…

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…

The values of the Riemann zeta function at odd positive integers, $\zeta(2n+1)$, are shown to admit a representation proportional to the finite-part of the divergent integral $\int_0^{\infty} t^{-2n-1} \operatorname{csch}t\,\mathrm{d}t$.…

I have made an ample study of one dimensional quantum oscillators, ranging from logarithmic to exponential potentials. I have found that the eigenvalues of the hamiltonian of the oscillator with the limiting (approachissimo) harmonic…

This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is…

Three geometric formulations of the Hamiltonian structure of the macroscopic Maxwell equations are given: one in terms of the double de Rham complex, one in terms of L2 duality, and one utilizing an abstract notion of duality. The final of…

Let $\mathbf{G}$ be a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring $\mathcal{O}$ of integers of a number field. We consider bivariate zeta functions of groups of the form $\mathbf{G}(\mathcal{O})$…

The Argand diagram is used to display some characteristics of the Riemann Zeta function. The zeros of the Zeta function on the complex plane give rise to an infinite sequence of closed loops, all passing through the origin of the diagram.…